{"id":"cb_question_bank:4e3ad962-acda-4c81-9041-1cf91af25ed2","source_id":"cb_question_bank","source_item_id":"4e3ad962-acda-4c81-9041-1cf91af25ed2","external_id":"72f977c3-5379-42b1-947f-d9fc4dbc27fb","ibn":null,"question_id":"2f51abc2","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"f left parenthesis x right parenthesis equals StartAbsoluteValue 59 minus 2 x EndAbsoluteValue The function f is defined by the given equation. For which of the following values of k does f left parenthesis k right parenthesis equals 3 k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"f left parenthesis x right parenthesis equals StartAbsoluteValue 59 minus 2 x EndAbsoluteValue\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mrow><mn>59</mn></mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math></p>\n<p style=\"text-align: left;\">The function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by the given equation. For which of the following values of <math alttext=\"k\"><mi>k</mi>\n</math> does <math alttext=\"f left parenthesis k right parenthesis equals 3 k\"><mi>f</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><mn>3</mn><mi>k</mi></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. The value of <math alttext=\"k\"><mi>k</mi>\n</math> for which <math alttext=\"f left parenthesis k right parenthesis equals 3 k\"><mi>f</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><mn>3</mn><mi>k</mi></math> can be found by substituting <math alttext=\"k\"><mi>k</mi>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"3 k\"><mrow>\n\t<mn>3</mn>\n\t<mi>k</mi>\n</mrow>\n</math> for&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the given equation, <math alttext=\"f left parenthesis x right parenthesis equals StartAbsoluteValue 59 minus 2 x EndAbsoluteValue\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math>, which yields <math alttext=\"3 k equals StartAbsoluteValue 59 minus 2 k EndAbsoluteValue\"><mn>3</mn><mi>k</mi><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow></mfenced></math>. For this equation to be true, either&nbsp;<math alttext=\"minus 3 k equals 59 minus 2 k\"><mo>-</mo><mn>3</mn><mi>k</mi><mo>=</mo><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></math> or <math alttext=\"3 k equals 59 minus 2 k\"><mn>3</mn><mi>k</mi><mo>=</mo><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></math>. Adding <math alttext=\"2 k\"><mrow>\n\t<mn>2</mn>\n\t<mi>k</mi>\n</mrow>\n</math> to both sides of the equation <math alttext=\"minus 3 k equals 59 minus 2 k\"><mo>-</mo><mn>3</mn><mi>k</mi><mo>=</mo><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></math> yields <math alttext=\"negative k equals 59\"><mo>-</mo><mi>k</mi><mo>=</mo><mn>59</mn></math>. Dividing both sides of this equation by <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math> yields <math alttext=\"k equals negative 59\"><mi>k</mi><mo>=</mo><mo>-</mo><mn>59</mn></math>. To check whether <math alttext=\"negative 59\"><mo>-</mo><mn>59</mn>\n</math> is the value of <math alttext=\"k\"><mi>k</mi>\n</math>, substituting <math alttext=\"negative 59\"><mo>-</mo><mn>59</mn>\n</math> for <math alttext=\"k\"><mi>k</mi>\n</math> in the equation <math alttext=\"3 k equals StartAbsoluteValue 59 minus 2 k EndAbsoluteValue\"><mn>3</mn><mi>k</mi><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow></mfenced></math> yields <math alttext=\"3 left parenthesis negative 59 right parenthesis equals StartAbsoluteValue 59 minus 2 left parenthesis negative 59 right parenthesis EndAbsoluteValue\"><mn>3</mn><mfenced><mrow><mo>-</mo><mn>59</mn></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>59</mn></mrow></mfenced></mrow></mfenced></math>, which is equivalent to <math alttext=\"negative 177 equals StartAbsoluteValue 177 EndAbsoluteValue\"><mo>-</mo><mn>177</mn><mo>=</mo><mfenced><mn>177</mn></mfenced></math>, or <math alttext=\"negative 177 equals 177\"><mo>-</mo><mn>177</mn><mo>=</mo><mn>177</mn></math>, which isn't a true statement. Therefore, <math alttext=\"negative 59\"><mo>-</mo><mn>59</mn>\n</math> isn't the value of <math alttext=\"k\"><mi>k</mi>\n</math>. Adding <math alttext=\"2 k\"><mrow>\n\t<mn>2</mn>\n\t<mi>k</mi>\n</mrow>\n</math> to both sides of the equation <math alttext=\"3 k equals 59 minus 2 k\"><mn>3</mn><mi>k</mi><mo>=</mo><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></math> yields <math alttext=\"5 k equals 59\"><mn>5</mn><mi>k</mi><mo>=</mo><mn>59</mn></math>. Dividing both sides of this equation by <math alttext=\"5\"><mn>5</mn>\n</math> yields <math alttext=\"k equals StartFraction 59 Over 5 EndFraction\"><mi>k</mi><mo>=</mo><mfrac><mn>59</mn><mn>5</mn></mfrac></math>. To check whether <math alttext=\"StartFraction 59 Over 5 EndFraction\"><mfrac>\n\t<mn>59</mn>\n\t<mn>5</mn>\n</mfrac>\n</math> is the value of <math alttext=\"k\"><mi>k</mi>\n</math>, substituting <math alttext=\"StartFraction 59 Over 5 EndFraction\"><mfrac>\n\t<mn>59</mn>\n\t<mn>5</mn>\n</mfrac>\n</math>&nbsp;for <math alttext=\"k\"><mi>k</mi>\n</math> in the equation&nbsp;<math alttext=\"3 k equals StartAbsoluteValue 59 minus 2 k EndAbsoluteValue\"><mn>3</mn><mi>k</mi><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow></mfenced></math> yields <math alttext=\"3 left parenthesis StartFraction 59 Over 5 EndFraction right parenthesis equals StartAbsoluteValue 59 minus 2 left parenthesis StartFraction 59 Over 5 EndFraction right parenthesis EndAbsoluteValue\"><mn>3</mn><mfenced><mfrac><mn>59</mn><mn>5</mn></mfrac></mfenced><mo>=</mo><mfenced><mrow><mn>59</mn><mo>-</mo><mn>2</mn><mfenced><mfrac><mn>59</mn><mn>5</mn></mfrac></mfenced></mrow></mfenced></math>, which is equivalent to <math alttext=\"StartFraction 177 Over 5 EndFraction equals StartAbsoluteValue StartFraction 177 Over 5 EndFraction EndAbsoluteValue\"><mfrac><mn>177</mn><mn>5</mn></mfrac><mo>=</mo><mfenced><mfrac><mn>177</mn><mn>5</mn></mfrac></mfenced></math>, or <math alttext=\"StartFraction 177 Over 5 EndFraction equals StartFraction 177 Over 5 EndFraction\"><mfrac><mn>177</mn><mn>5</mn></mfrac><mo>=</mo><mfrac><mn>177</mn><mn>5</mn></mfrac></math>, which is a true statement. Therefore, the value of <math alttext=\"k\"><mi>k</mi>\n</math> for which <math alttext=\"f left parenthesis k right parenthesis equals 3 k\"><mi>f</mi><mfenced><mi>k</mi></mfenced><mo>=</mo><mn>3</mn><mi>k</mi></math>&nbsp;is <math alttext=\"StartFraction 59 Over 5 EndFraction\"><mfrac><mn>59</mn><mn>5</mn></mfrac></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.829000+00:00","source_updated_at":"2023-08-02T20:25:59.829000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"StartFraction 59 Over 5 EndFraction\"><mfrac>\n\t<mn>59</mn>\n\t<mn>5</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction 59 Over 2 EndFraction\"><mfrac>\n\t<mn>59</mn>\n\t<mn>2</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction 177 Over 5 EndFraction\"><mfrac>\n\t<mn>177</mn>\n\t<mn>5</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"59\"><mn>59</mn>\n</math></p>","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}